
We first assemble all the \(D\) -bubbles of \({\mathcal {G}}\)  into a set \(A\) :
\(A = \Bigl \lbrace  {\mathcal {B}}^{\widehat{i}}_{(\rho )}\colon i \in \lbrace  0,\dots , D \rbrace ,\; \rho \;\textrm {unrestricted} \Bigr \rbrace  .\) 

For each \((D+1-d)\) -bubble within \({\mathcal {G}}\) , \(d\in \lbrace 1,\dots , D+1\rbrace \) , we form the following subset of \(A\) :
\(\sigma ^{ {\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) } } =\Bigl \lbrace  {\mathcal {B}}^{\widehat{i}_k}_{(\rho )} \colon {\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) }\subset {\mathcal {B}}^{\widehat{i}_k}_{(\rho )} , \; k\in \lbrace 1,\dots , d\rbrace  \Bigr \rbrace  \subset A .\) 
In fact, for a given \((D+1-d)\) -bubble \({\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) }\) , there is, for each \(k\in \lbrace 1,\dots ,d\rbrace \) , a unique \(D\) -bubble \({\mathcal {B}}^{\widehat{i}_k}_{(\rho )} \supset {\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) }\) . This is the maximal connected component (in \({\mathcal {G}}\) ) obtained by starting from \({\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) }\)  and adding edges of all colors except \(i_k\) . Thus, the cardinality of \(\sigma ^{ {\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) } }\)  is \(d\) .

As a result, any subset \(\tau \in \sigma ^{ {\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) } } \)  is indexed by a choice of subset \(S \subset \lbrace  1,\dots , d \rbrace  \) :
\(\tau = \Bigr \lbrace  {\mathcal {B}}^{\widehat{i}_k}_{(\rho )} \,|\, {\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) }\subset {\mathcal {B}}^{\widehat{i}_k}_{(\rho )} ,\; k \in \lbrace  1,\dots , d \rbrace  \setminus S \Bigl \rbrace  .\) 
So \(\tau = \sigma ^{{\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_{d-|S|}}_{ (\xi ) }}\) , where \({\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_{d-|S|}}_{ (\xi ) }\)  is the unique subgraph obtained by adding the colors \(i_s\) , \(s\in S\) , to the subgraph \({\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_d}_{ (\kappa ) }\) .

The final crucial detail is that the sets \(\sigma ^{{\mathcal {B}}^{ \widehat{i}_1 \widehat{i}_2\dots \widehat{i}_{d}}_{ (\kappa ) }}\)  are the \((d-1)\) -simplices of a finite abstract simplicial complex, defined as
\(\Delta = \Big \lbrace  \sigma ^{ {\mathcal {B}}^{\hat{i}_1\hat{i}_2\dots \hat{i}_d}_{ (\kappa ) } } \, | \,{\mathcal {B}}^{\hat{i}_1\hat{i}_2\dots \hat{i}_d}_{ (\kappa ) } \subset {\mathcal {G}}\Big \rbrace .\) 
It is straightforward to verify the defining property of an abstract simplicial complex: that for all \(\sigma \in \Delta \)  and \(\tau \subset \sigma \) , then \(\tau \in \Delta \) .
The cardinality of \(\sigma \in \Delta \)  is \(d\)  (it corresponds to a \((D+1-d)\) -bubble) and so its dimension is \(d-1\) .

In fact, since \(\Delta \)  is non-branching, strongly connected and pure, it is a \(D\) -dimensional simplicial pseudomanifold [1]}.

